What is the derivative of (4-t)(2+t^2)^{-1} ?

The derivative of (4-t)(2+t^2)^{-1} is (t^2-8t-2)/((t^2+2)^2)

What is the first derivative of (4-t)(2+t^2)^{-1} ?

The first derivative of (4-t)(2+t^2)^{-1} is (t^2-8t-2)/((t^2+2)^2)

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derivative of

(4 - t) (2 + t^{2})^{- 1}

\frac{t ^{2} - 8 t - 2}{( t ^{2} + 2 ) ^{2}}

Solution steps

\frac{d}{d t} ((4 - t) (2 + t^{2})^{- 1})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = 4 - t, g = (2 + t^{2})^{- 1}

= \frac{d}{d t} (4 - t) (2 + t^{2})^{- 1} + \frac{d}{d t} ((2 + t^{2})^{- 1}) (4 - t)

\frac{d}{d t} (4 - t) = - 1

\frac{d}{d t} ((2 + t^{2})^{- 1}) = - \frac{2 t}{( 2 + t ^{2} ) ^{2}}

= (- 1) (2 + t^{2})^{- 1} + (- \frac{2 t}{( 2 + t ^{2} ) ^{2}}) (4 - t)

Simplify

(- 1) (2 + t^{2})^{- 1} + (- \frac{2 t}{( 2 + t ^{2} ) ^{2}}) (4 - t) : \frac{t ^{2} - 8 t - 2}{( t ^{2} + 2 ) ^{2}}

= \frac{t ^{2} - 8 t - 2}{( t ^{2} + 2 ) ^{2}}

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What is the derivative of (4-t)(2+t^2)^{-1} ?
The derivative of (4-t)(2+t^2)^{-1} is (t^2-8t-2)/((t^2+2)^2)
What is the first derivative of (4-t)(2+t^2)^{-1} ?
The first derivative of (4-t)(2+t^2)^{-1} is (t^2-8t-2)/((t^2+2)^2)